It is well known that if two independent identically distributed random variables are Gaussian, then their sum and difference are also independent. It turns out that only Gaussian random variables have such property. This statement, known as the famous Kac-Bernstein theorem, is a typical example of...

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Bibliographic Details

Main Author:	Felʹdman, G. M. (Author)
Format:	Electronic eBook
Language:	English
Published:	Providence, Rhode Island American Mathematical Society [2023]
Series:	Mathematical surveys and monographs Volume 273
Subjects:	Locally compact Abelian groups Distribution (Probability theory) Probability theory and stochastic processes > Probability theory on algebraic and topological structures > Probability measures on groups or semigroups, Fourier transforms, factorization Statistics > Distribution theory > Characterization and structure theory Abstract harmonic analysis > Abstract harmonic analysis > Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups Abstract harmonic analysis > Abstract harmonic analysis > Positive definite functions on groups, semigroups, etc Difference and functional equations > Functional equations and inequalities > Equations for functions with more general domains and/or ranges
Online Access:	Volltext
Summary:	It is well known that if two independent identically distributed random variables are Gaussian, then their sum and difference are also independent. It turns out that only Gaussian random variables have such property. This statement, known as the famous Kac-Bernstein theorem, is a typical example of a so-called characterization theorem. Characterization theorems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions of these random variables. The first results in this area are associated with famous 20th century mathematicians such as G. Pólya, M. Kac, S. N. Bernstein, and Yu. V. Linnik. By now, the corresponding theory on the real line has basically been constructed. The problem of extending the classical characterization theorems to various algebraic structures has been actively studied in recent decades. The purpose of this book is to provide a comprehensive and self-contained overview of the current state of the theory of characterization problems on locally compact Abelian groups. The book will be useful to everyone with some familiarity of abstract harmonic analysis who is interested in probability distributions and functional equations on groups.
Item Description:	Includes bibliographical references and index
Physical Description:	1 Online-Ressource (ix, 240 Seiten)
ISBN:	9781470473266 9781470472955
DOI:	10.1090/surv/273